Really the value of Bitcoin is based on the velocity of money, and speculation. The latter is just a matter of supply and demand, it's the former that's more interesting. Suppose Alice, in the USA, wants to transfer value to Bob, who lives in Germany, using Bitcoins. The actual exchange of value is going to look like this:
- Alice gives Charlie, who runs an exchange, some US dollars. Charlie gives Alice some Bitcoins.
- Alice sends her Bitcoins to Bob.
- Bob gives Dan, who also runs an exchange, his new Bitcoins, and Dan gives Bob Euros.
Now if this goes on over and over again, Charlie is going to be short of Bitcoins, and Dan short of Euros. So on Charlie's exchange the price of Bitcoin relative to the US dollar will rise, and on Dan's exchange, it'll fall. Enter Anna, are arbitrage trader:
- Anna sells US dollars on the Forex market, buying Euros.
- Now Anna wires those Euros to Dan's bank account in Germany, and Dan gives Anna Bitcoins
- Anna then gives Charlie Bitcoins, getting US dollars back.
Why didn't Alice just wire Euro's? Well, bank wires have high fixed fees, so Anna can amortize the fees over one huge wire transfer in a way that Alice can't. Also, Anna doesn't care if either side know who she is, but Alice might.
The key thing is that all these steps take time, which means Bitcoins are tied up in the system, reducing the supply. So by basic supply and demand, the price goes up, based on how useful it is to use Bitcoins to transfer value between people, and how fast the whole process can happen. Some people find them very useful to do this, especially pseudo-anonymously, notably the drug trading site Silkroad. Note how curiously a more efficient Bitcoin market, in terms of fiat conversion, will lead to a lower Bitcoin price.
The key thing is that Alice and Bob don't actually care that much what the price of Bitcoin relative to other currencies is. They just care that the price doesn't change fast enough that one side or the other gets ripped off during the time it takes to complete transaction. The faster those transactions happen, the less of an issue this is. If either side can quickly buy and sell their coins, something made easier by things like BitInstant's new paycard, they don't have to worry about exchange rate volatility as much. And speaking of, while people in the US don't have this problem, ask someone in, say, Iceland, with its tiny economy and its own currency, how stable the value of the Krona is relative to other currencies...
As for what happens if cash systems collapse... frankly if that's true, chances are something is seriously wrong with society. If the internet is still functioning Bitcoin might still be working well enough as a technological project that your coins are going to be worth something. If the internet isn't work, well, you're coins are going to be worthless because people can't transfer them. Of course, if you're in such bad straits that what you really want is goats and chickens, you might also quickly find out that you can't eat gold...